Lie symmetries and differential Galois groups of linear equations

Oudshoorn, W.; van der Put, M.

doi:10.1090/S0025-5718-01-01397-7

Lie symmetries and differential Galois groups of linear equations
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by W. R. Oudshoorn and M. van der Put PDF

Math. Comp. 71 (2002), 349-361 Request permission

Abstract:

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differential equation is presented.

References

Irving Kaplansky, An introduction to differential algebra, 2nd ed., Publications de l’Institut de Mathématique de l’Université de Nancago, No. V, Hermann, Paris, 1976. MR 0460303
Jorge Krause and Louis Michel, Équations différentielles linéaires d’ordre $n>2$ ayant une algèbre de Lie de symétrie de dimension $n+4$, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 18, 905–910 (French, with English summary). MR 978467
S. Lie, Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen, Bearbeitet und herausgegeben von Dr. G. Scheffers, Teubner, Leipzig, 1893.
S. Lie, Klassifikation und Integration von gewöhnlicher Differentialgleichungen zwischen $x,y$, die eine Gruppe von Transformationen gestetten I, Math. Ann. 22 (1888), 213–253.
F.M. Mahomed and P.G.L. Leach, Symmetry Lie algebras of $n$th order ordinary differential equations, J. Math. Anal. Appl. 151 (1990), 80–107.
Peter J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR 836734, DOI 10.1007/978-1-4684-0274-2
Peter J. Olver, Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995. MR 1337276, DOI 10.1017/CBO9780511609565
Marius van der Put, Galois theory of differential equations, algebraic groups and Lie algebras, J. Symbolic Comput. 28 (1999), no. 4-5, 441–472. Differential algebra and differential equations. MR 1731933, DOI 10.1006/jsco.1999.0310
Hans Stephani, Differential equations, Cambridge University Press, Cambridge, 1989. Their solution using symmetries. MR 1041800